The Bar-Radical of a Class of Almost Alternative Baric Algebras

In this article we prove that, if (U, ω) is a finite dimensional baric algebra of (γ, δ) type over a field F of characteristic ≠ 2,3,5 such that γ² ? δ² + δ = 1 and δ ≠ 0,1, then rad(U) = R(U) ∩ (bar(U))², where R(U) is the nilradical (maximal nil ideal) of U.     

The Double Ringel-Hall Algebras of Valued Quivers

This paper is devoted to the study of the structure of the double Ringel-Hall algebra D(Λ) for an infinite dimensional hereditary algebra A, which is given by a valued quiverΓover a finite field, and also to the study of the relations of D(Λ)-modules with representations of valued quiverΓ.     

The Composition Structure of Alternative and Malcev Algebras

We construct a polynomial of degree 5 from the associative nucleus (kernel) of the free alternative algebra. We show that this polynomial is of minimal degree. Using this polynomial, we obtain decompositions of the varieties of alternative and Malcev algebras.     

Flat Nonunimodular Lorentzian Lie Algebras

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (?, +,…, +) such that left multiplications are skew-symmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4.     

Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Since 1976, it is known from the paper by V. P. Belkin that the variety RA₂ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains nonfinitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann RA₂-algebra of finite rank r over a field ?, for char(?) ≠ 2, is Spechtian iff r = 1. We construct a nonfinitely based variety 𝔐 generated by the Grassmann 𝒱-algebra of rank 2 of certain finitely based subvariety 𝒱 ? RA₂ over a field ?, for char(?) ≠ 2, 3, such that 𝔐 can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.     

Structures of Not-finitely Graded Lie Algebras Related to Generalized Virasoro Algebras

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.     

On Cartan Invariants and Blocks of Zassenhaus Algebras #

In this article, we determine the Cartan invariants for Zassenhaus algebras W(1,n). This is done by reducing representations of generalized restricted Cartan type Lie algebra W(1,n) to representations of restricted Lie algebras W(1,1) and of ± b𝔰 ± b𝔩(2), and then extending Feldvoss-Nakano's argument on W(1,1) to the case W(1,n).     

Multiplicative Semiprimeness of Skew Lie Algebras

Abstract

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z _{[K, K]} denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z _{[K, K]} is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z _{[K, K]} is prime, whenever [K, K]/Z _{[K, K]} is prime. We will obtain similar results for the Lie algebra K/Z _K whenever the base field has characteristic zero.     

On Right Alternative Superalgebras

Simple right alternative superalgebras which have a simple algebra as even part and, as odd part, an irreducible bimodule over the even part are investigated. Under these conditions, superalgebras with one dimensional even part are classified, as well as superalgebras having M ₂(F) as even part and a unital irreducible bimodule over M ₂(F) of dimension less than or equal to 6 as odd part. It is shown that there is only a unique non alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

弱Hopf代数上的弱Alternative Doi-Hopf模

该文首先引入了弱Hopf代数上的弱Alternative Doi-Hopf模,然后构造了从弱Alternative Doi-Hopf模范畴到模范畴(余模范畴)忘却函子的伴随函子.     

Twisted Comodule Algebras and the Generalized Quantum Double

In this article we show that the notion of twisted comodule algebras is a generalization of Drinfeld's cocycle twisting construction of bialgebras or Hopf algebras, and that several other twisting constructions, such as Ferrer's twisting product and Lu's one-sided twisting, are also special examples of twisted comodule algebras. Some general properties on twisted comodule algebras are investigated, and applications to the generalized quantum double are given.     

Structures of Generalized Loop Virasoro Algebras

In this article, we study the structure theory of a class of generalized loop Virasoro algebras. In particular, the derivation algebras, the automorphism groups, and the second cohomology groups of generalized loop centerless Virasoro algebras are determined.     

Embedding of Lie Algebras in Isogeneralized Structurable Algebras

In a previous paper, the first-named author introduced generalized structurable algebras, while the second-named author introduced the isotopies of Lie algebras. In this paper, we combine the two analyses, submit the notion of isogeneralized structural algebras, and show that they include Lie algebras, all their axiom-preserving generalizations of graded, supersymmetric or isotopic type, as well as numerous other algebras.     

Construction of Nilpotent Jordan Algebras Over any Arbitrary Field

The paper is devoted to the study of annihilator extensions of Jordan algebras and suggests new approach to classify nilpotent Jordan algebras, which is analogous to the Skjelbred–Sund method for classifying nilpotent Lie algebras [2 de Graaf, W. (2007). Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. J. Algebra 309:640–653.[Crossref], [Web of Science ®] , [Google Scholar], 4 Gong, M.-P. (1998). Clasification of Nilpotent Lie Algebras of Dimension 7 [Ph.D]. Ontario, Canada: University of Waterloo. [Google Scholar], 15 Skjelbred, T., Sund, T. (1978). Sur la classification des algèbres de Lie nilpotentes. C. R. Acad. Sci. Paris Sér. A-B 286:241–242. [Google Scholar]]. Subsequently, we have classified nilpotent Jordan algebras of dimension up to four.     

Quotients of Koszul Algebras and 2-d-Determined Algebras

Vatne [13 Vatne , J. E. ( 2012 ). Quotients of Koszul algebras with almost linear resolution. Preprint, arXiv:1103.3572 . [Google Scholar]] and Green and Marcos [9 Green , E. L. , Marcos , E. N. (2011). d-Koszul algebras, 2-d-determined algebras and 2-d-Koszul algebras. J. Pure Appl. Algebra 215(4):439–449.[Crossref], [Web of Science ®] , [Google Scholar]] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.     

Polynomial Identities of RA and RA2 Loop Algebras

Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z _F over F. For RA2 loops of type M(Dih(A), ^?1,g ₀), we prove that the loop algebra is in the variety generated by the algebra  ₃ which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, ^?1,g ₀), where G is a non-Abelian group of exponent 4 having exactly 2 squares.